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%I #17 Jun 30 2016 02:02:08
%S 1,1,2,1,2,1,2,3,4,1,3,4,5,1,2,4,2,3,4,5,7,2,3,5,6,7,8,1,3,6,7,8,9,1,
%T 2,4,7,3,4,5,7,8,11,1,4,5,6,8,9,11,12,2,5,6,9,10,11,12,13,1,3,6,10,11,
%U 12,13,14,1,2,4,7,11,3,5,7,8,11,12,16,2,6,7
%N Table read by rows: row n gives all numbers less than n in the same row, column, or diagonal as n in the natural numbers read by antidiagonals.
%H Peter Kagey, <a href="/A274080/b274080.txt">Table of n, a(n) for n = 1..10000</a>
%e A000027 read by antidiagonals is:
%e 1 2 4 7
%e 3 5 8
%e 6 9
%e ...
%e Thus:
%e Row 1: []
%e Row 2: [1]
%e Row 3: [1, 2]
%e Row 4: [1, 2]
%e Row 5: [1, 2, 3, 4]
%e Row 6: [1, 3, 4, 5]
%e Row 7: [1, 2, 4]
%e Row 8: [2, 3, 4, 5, 7]
%e Row 9: [2, 3, 5, 6, 7, 8]
%t nn = 18; t = Table[(n^2 - n)/2 + Accumulate@ Range[n - 1, Ceiling[(Sqrt[9 + 8 nn] - 3)/2]] + 1, {n, Ceiling[(Sqrt[9 + 8 nn] - 3)/2] + 1}]; Table[Function[a, Function[p, Most@ Union@ Flatten@ {Map[a[[#1, #2]] & @@ # &, Most@ NestWhileList[# - 1 &, First@ p, ! MemberQ[#, 0] &]], Range[SelectFirst[Reverse@ Join[{0}, First@ t], n >= # &], n - 1], Transpose[a][[ p[[1, 2]] ]], a[[ p[[1, 1]] ]]}]@ Position[a, n]]@ Array[t[[#1, #2]] &, First@ Position[t, n]], {n, nn}] // Flatten (* _Michael De Vlieger_, Jun 29 2016, Version 10 *)
%o (Haskell)
%o import Data.List (sort, nub)
%o a274080 n = a274080_list !! (n - 1)
%o a274080_list = concatMap a274080_row [1..]
%o a274080_tabf = map a274080_row [1..]
%o a274080_row n = nub $ sort $ concatMap (\f -> f n) [a274079_row, a273825_row, a273824_row, a273823_row]
%Y Cf. A000027, A269526, A273823, A273824, A273825, A274079.
%K nonn,tabf,look
%O 1,3
%A _Peter Kagey_, Jun 09 2016
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